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arXiv · 1907.12633

Tracer turbulence: the Batchelor-Howells-Townsend spectrum revisited

Michael S. Jolly·Djoko Wirosoetisno
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Abstract

Given a velocity field $u(x,t)$, we consider the evolution of a passive tracer $\theta$ governed by $\partial_t\theta + u\cdot\nabla\theta = \Delta\theta + g$ with time-independent source $g(x)$. When $\|u\|$ is small, Batchelor, Howells and Townsend (1959, J.\ Fluid Mech.\ 5:134) predicted that the tracer spectrum scales as $|\theta_k|^2\propto|k|^{-4}|u_k|^2$. In this paper, we prove that this scaling does indeed hold for large $|k|$, in a probabilistic sense, for random synthetic two-dimensional incompressible velocity fields $u(x,t)$ with given energy spectra. We also propose an asymptotic correction factor to the BHT scaling arising from the time-dependence of $u$.

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Michael S. Jolly, Djoko Wirosoetisno. 2019-07-29. Tracer turbulence: the Batchelor-Howells-Townsend spectrum revisited. https://doi.org/10.1007/s00021-019-0478-6

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